(a) max |R^+|/eps_+ = 0.499686 at (r, theta) = (400, 0.0); max |R^-|/eps_- = 0.338344 at (5, 1.5707963267948966)  (429 grid points, r in [5, 400])
(b) variation of 1/t along the admissible paths (quadrature vs closed form):
   r=5, theta=0: V_+ = 0.2000015708 (1/r = 0.2); V_- (X=1e4) = 0.2000570511 <= theta/(r sin theta) + pi/(x+X) = 0.2003140023; limit theta/(r sin theta) = 0.2 <= pi/(2r) = 0.3141592654
   r=5, theta=0.3: V_+ = 0.2000012708 (1/r = 0.2); V_- (X=1e4) = 0.2030888689 <= theta/(r sin theta) + pi/(x+X) = 0.203345811; limit theta/(r sin theta) = 0.2030318017 <= pi/(2r) = 0.3141592654
   r=7, theta=0.9: V_+ = 0.1428578137 (1/r = 0.1428571429); V_- (X=1e4) = 0.1641921942 <= theta/(r sin theta) + pi/(x+X) = 0.1644491072; limit theta/(r sin theta) = 0.1641350846 <= pi/(2r) = 0.2243994753
   r=60, theta=0.785398: V_+ = 0.01666745206 (1/r = 0.01666666667); V_- (X=1e4) = 0.01856927201 <= theta/(r sin theta) + pi/(x+X) = 0.01882484427; limit theta/(r sin theta) = 0.01851201224 <= pi/(2r) = 0.02617993878
   r=60, theta=1.5708: V_+ = 0.01666666667 (1/r = 0.01666666667); V_- (X=1e4) = 0.02623760294 <= theta/(r sin theta) + pi/(x+X) = 0.02649408138; limit theta/(r sin theta) = 0.02617992211 <= pi/(2r) = 0.02617993878
   r=10, theta=1.5708: V_+ = 0.1 (1/r = 0.1); V_- (X=1e4) = 0.1571368123 <= theta/(r sin theta) + pi/(x+X) = 0.1573937919; limit theta/(r sin theta) = 0.1570796327 <= pi/(2r) = 0.1570796327
   Volterra kernel |(1 - e^(2i(t-z)))/(2i)| over 2000 random pairs with Im t >= Im z: max = 0.964843 (<= 1)
(c) limit 1/8 - i/(2 pi) = (0.125 - 0.1591549431j), |.| = 0.2023741483
   m=10: (z_m - F_m) F_m = (0.12629019 - 0.15782577j), (2 pi m)|z_m - F_m| = 0.19949126
   m=20: (z_m - F_m) F_m = (0.12564117 - 0.15847827j), (2 pi m)|z_m - F_m| = 0.20093491
   m=40: (z_m - F_m) F_m = (0.1253191 - 0.15881313j), (2 pi m)|z_m - F_m| = 0.20165705
   m=100: (z_m - F_m) F_m = (0.12512715 - 0.15901726j), (2 pi m)|z_m - F_m| = 0.20208856
   m=1000: (z_m - F_m) F_m = (0.12501267 - 0.15914111j), (2 pi m)|z_m - F_m| = 0.20234574
   m=5000: (z_m - F_m) F_m = (0.12500253 - 0.15915217j), (2 pi m)|z_m - F_m| = 0.20236847
   m=20000: (z_m - F_m) F_m = (0.12500063 - 0.15915425j), (2 pi m)|z_m - F_m| = 0.20237273
   max of (2 pi m)|z_m - F_m| over the sample = 0.20237273
   gaps near m = 20000: arg z_19999 - arg z_20000 = 2.50265e-9, arg z_20000 - arg z_20001 = 2.50241e-9; 0.3196/m^2 = 7.99e-10
NUMERIC (non-rigorous) CHECKS DONE
